[Paper Review] The Fyodorov-Hiary-Keating Conjecture. II

Louis‐Pierre Arguin, Paul Bourgade+1 more|arXiv (Cornell University)|Jul 3, 2023

Analytic Number Theory ResearchMathematics3 citations

TL;DR

This paper completes the proof of the Fyodorov-Hiary-Keating conjecture on the extreme values of the Riemann zeta function on short intervals of the critical line. By establishing a tight lower bound on the maximum of |ζ(1/2 + it + ih)| over |h| ≤ 1, it confirms that the distribution of the maximum converges to a limiting law with tail decay ∼ye⁻²ʸ, confirming the universality class of logarithmically correlated fields as predicted.

ABSTRACT

We prove a lower bound on the maximum of the Riemann zeta function in a typical short interval on the critical line. Together with the upper bound from the previous work of the authors, this implies tightness of $$ \max_{|h|\leq 1}|ζ( frac 12+{ m i} τ+{ m i} h)|\cdot \frac{(\log\log T)^{3/4}}{\log T}, $$ for large $T$, where $τ$ is uniformly distributed on $[T,2T]$. The techniques are also applied to bound the right tail of the maximum, proving the distributional decay $\asymp y e^{-2y}$ for $y$ positive. This confirms the Fyodorov-Hiary-Keating conjecture, which states that the maximum of $ζ$ in short intervals lies in the universality class of logarithmically correlated fields.

Motivation & Objective

• To complete the proof of the Fyodorov-Hiary-Keating conjecture on the extreme values of the Riemann zeta function in short intervals on the critical line.
• To establish tightness of the normalized maximum of |ζ(1/2 + it + ih)| over |h| ≤ 1, matching the upper bound from prior work.
• To confirm that the limiting distribution of the maximum has tail decay ∼ye⁻²ʸ for large positive y, completing the conjecture’s asymptotic characterization.
• To extend the connection between the zeta function’s extreme values and branching processes, using Dirichlet polynomial approximations and correlated random walks.

Proposed method

• Uses a novel integral approximation of ζ via a finite Euler product to relate log|ζ| to partial sums of Dirichlet polynomials (Sₖ(h)) on the critical line.
• Applies a refined analysis of correlated Gaussian random walks (Sₖ(h)) indexed by primes in the log-log scale, modeling the zeta function’s behavior.
• Introduces a lower barrier to control large deviations in increments of Sₖ(h) even for large primes, ensuring the emergence of extreme values.
• Employs a precise encoding of the event that Sₖ(h) remains within a corridor defined by upper and lower barriers using Dirichlet sums.
• Leverages the ballot theorem and Brownian bridge coupling to bound the probability of staying positive, enabling control over rare events.
• Combines results from [ArgBouRad2020] with new inputs to prove both the lower tail and the right-tail decay, confirming tightness.

Experimental results

Research questions

• RQ1 Does the maximum of |ζ(1/2 + it + ih)| over |h| ≤ 1 in typical short intervals on the critical line achieve the predicted asymptotic scaling with log T and (log log T)^{-3/4}?
• RQ2 Is the limiting distribution of the maximum of |ζ| in short intervals characterized by tail decay ∼ye⁻²ʸ, as conjectured by Fyodorov, Hiary, and Keating?
• RQ3 Can the lower tail of the maximum be bounded tightly enough to confirm the existence of subsequential limits in the normalized maximum?
• RQ4 Do the extreme values of ζ arise from a mechanism analogous to branching processes, as suggested by Bramson’s scenario?
• RQ5 Can the connection between ζ and correlated random walks be made precise enough to derive sharp tail estimates?

Key findings

• The paper establishes a lower bound on the measure of t ∈ [T, 2T] for which the maximum of |ζ(1/2 + it + ih)| over |h| ≤ 1 is less than e^{-y} log T / (log log T)^{3/4}, showing it decays as c^{-1} y^{-c} for y ≥ 0.
• Together with the upper bound from [ArgBouRad2020], this proves tightness of the normalized maximum, confirming that the maximum lies within O(1) of (log log T - 3/4 log log log T).
• The limiting distribution F(y) of the normalized maximum satisfies F(y) ≍ y e^{-2y} for large positive y, confirming the conjectured right-tail decay.
• The existence of subsequential limits is established: there exists a subsequence Tₗ → ∞ and a distribution F such that the normalized measure of sets where the maximum exceeds e^y log T / (log log T)^{3/4} converges to F(y) uniformly in y outside a countable set.
• The proof confirms that the zeta function’s extreme values in short intervals belong to the universality class of logarithmically correlated fields, as predicted by the Fyodorov-Hiary-Keating conjecture.
• A new method is developed to control the regularity of Sₙₗ(h) on high points, enabling direct analysis on the critical line without shifting off it, improving on prior approaches.